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  <a class="pure-menu-link nav1" onclick="animateByNav()" href="#_1">第三部分 概率论</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#1">第1讲 随机事件和概率</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_1">1. 概率基本公式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2">2. 条件概率</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4">4. 全概率公式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5">5. 贝叶斯公式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#6">6. 超几何分布</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#7">7. 独立性</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_2">（1）两个事件的独立的定义</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_1">（2）两个事件独立可以得到的结论</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#2_2">第2讲 一维随机变量及其分布</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3">3. 反函数的概率密度函数</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#4_1">第4讲 多维随机变量及其分布</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5_1">5. 常见组合随机变量的概率密度函数</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4_2">（4）叠加正态分布</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#5_2">（5）二维正态分布</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#6_1">第6讲 数字特征</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_3">1. 数学期望（均值）</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_3">2. 方差</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_1">3. 边缘概率密度</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_3">4. 协方差</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5_3">5. 相关系数</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#7_1">第7讲 大数定理与中心极限定理</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_4">1. 依概率收敛</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_4">2. 大数定理</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_5">（1）切比雪夫大数定律</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2bernoulli">（2）伯努利(Bernoulli)大数定律</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3khinchine">（3）辛钦(Khinchine)大数定律</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4_4">（4）大数定理的结论</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_2">3. 中心极限定理</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1-levy-lindberg">（1）列维-林德伯格(Levy-Lindberg)定理</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2-de-moivre-laplace">（2）棣莫弗-拉普拉斯(De Moivre-Laplace)定理</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#8">第8讲 统计量及其分布</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_6">1. 统计量</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_7">（1）样本均值</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_5">（2）样本方差</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3_3">（3）样本标准差</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4-k">（4）样本 <script type="math/tex"> k </script> 阶原点距</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#5-k">（5）样本 <script type="math/tex"> k </script> 阶中心距</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_6">2. 统计量四大分布</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1nmusigma">（1）<script type="math/tex">N(\mu,\sigma)</script> 正态分布</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2-chi2">（2） <script type="math/tex"> \chi^2 </script> 分布</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3-t">（3） <script type="math/tex"> t </script> 学生分布</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4-f">（4） <script type="math/tex"> F </script> 分布</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#9">第9讲 参数估计</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_8">1. 矩估计</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_7">2. 最大似然估计</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_4">3. 区间估计</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_5">4. 正态总体均值与方差的区间估计</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5_4">5. 估计量的评选标准</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_9">（1）无偏性</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_8">（2）有效性</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3_5">（3）相合性</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#6_2">6. 假设检验</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#_2"></a>
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  <h1 id="数学-概率论" class="content-subhead">数学-概率论</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
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    <h1 id="_1">第三部分 概率论</h1>
<h2 id="1">第1讲 随机事件和概率</h2>
<h3 id="1_1">1. 概率基本公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(A\overline B) =&\ P(A-B) \\[1ex]
=&\ P(A)-P(AB) \\[3ex]
P(A+B) =&\ P(A)+P(B)-P(AB) \\[3ex]
P(A+B+C) =&\ P(A)+P(B)+P(C) \\[1ex]
&-P(AB)-P(AC)-P(BC)+P(ABC) \\[1ex]
\end{split}\end{equation}
</script>
</p>
<h3 id="2">2. 条件概率</h3>
<p>
<script type="math/tex; mode=display">
P(A|B) = \cfrac{P(AB)}{P(B)}
</script>
</p>
<h3 id="3-pab">3. 求 <script type="math/tex"> P(AB) </script>
</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(AB) &= P(B)P(A|B) = P(A)P(B|A) \ &\leftarrow 条件概率 P(A|B) = \cfrac{P(AB)}{P(B)}\\[2ex]
&= P(A) + P(B) - P(A+B) \ &\leftarrow P(A+B) = P(A)+P(B)-P(AB)\\[1ex]
&= P(A) - P(A\overline B) \ &\leftarrow P(A\overline B) = P(A)-P(AB)\\[1ex]
\end{split}\end{equation}
</script>
</p>
<h3 id="4">4. 全概率公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\sum_{i=1}^nP(A_i) &= 1 \\[1ex]
A_1, A_2, ..., A_i &构成一个完备事件，且都有正概率\\[1em]
P(B) &= \sum_{i=1}^nP(A_i)P(B|A_i)
\end{split}\end{equation}
</script>
</p>
<h3 id="5">5. 贝叶斯公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(A_j|B) &= \cfrac{P(A_jB)}{P(B)} \\[1ex]
&= \cfrac{P(A_j)P(B|A_j)}{\sum_{i=1}^nP(A_i)P(B|A_i)} = \cfrac{条件概率}{全概率}
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-概率论.assets/截屏2020-12-19 22.58.17.jpg" alt="截屏2020-12-19 22.58.17" style="zoom:33%;" /></p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-概率论.assets/截屏2020-12-19 23.09.39.jpg" alt="截屏2020-12-19 23.09.39" style="zoom:33%;" /></p>
<h3 id="6">6. 超几何分布</h3>
<p>
<script type="math/tex; mode=display">
p(k) = \cfrac{C_D^k * C_{N-D}^{n-k}}{C_N^n}
</script>
</p>
<h3 id="7">7. 独立性</h3>
<h4 id="1_2">（1）两个事件的独立的定义</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(AB)=P(A)P(B)
&⟺A与B相互独立 \\[1ex]
&⟺A与\overline B相互独立 \\[1ex]
&⟺\overline A与B相互独立 \\[1ex]
&⟺\overline A与\overline B相互独立
\end{split}\end{equation}
</script>
<br />
多个事件的独立性：</p>
<p>
<script type="math/tex; mode=display">
若 A,B,C,D 相互独立，则 \\[2ex]
(1)\ \ AB 与 CD 相互独立 \\[2ex]
(2)\ \ A 与 BC-D 相互独立(不包含对方事件)
</script>
</p>
<h4 id="2_1">（2）两个事件独立可以得到的结论</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
若A,B相互独立,并且0<P(B)\ \ \ \ \ \ \ \ &⟺P(A|B)=P(A) \\[2ex]
若A,B相互独立,并且0<P(B)<1&⟺P(A|B)=P(A|\overline{B}) \\[1ex]
&⟺P(A|B)+P(\overline{A}|\overline{B})=1
\end{split}\end{equation}
</script>
</p>
<p>其他结论<br />
<script type="math/tex; mode=display">
若0<P(A)<1且0<P(B)<1 \\[1ex]
且A,B互斥或者存在包含关系 \\[1ex]
则A,B一定\underline{不独立} \\[1ex]
若P(A)=0或1,\ \ \ 则A与任意事件相互独立
</script>
</p>
<h2 id="2_2">第2讲 一维随机变量及其分布</h2>
<h3 id="1-fx">1. 概率密度函数 <script type="math/tex">f(x)</script>
</h3>
<p>
<script type="math/tex; mode=display">
f(x)
</script>
</p>
<p>特性<br />
<script type="math/tex; mode=display">
f(x)\ge0,\ 且\ \int_{-\infty}^{+\infty} f(x)dx=1
</script>
</p>
<h3 id="2-fx">2. 概率分布函数 <script type="math/tex">F(x)</script>
</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
F(x)&=P\{X\le x\} \\[2ex]
&=\int_{-\infty}^x f(x)dx
\end{split}\end{equation}
</script>
</p>
<p>特性<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&F(x)单调不减,右连续 \\[2ex]
&F(-\infty)=0 \\[1ex]
&F(+\infty)=1
\end{split}\end{equation}
</script>
<br />
分布函数和概率的关系<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P\{X\le x\}&=F(x) \\[1ex]
P\{X\gt x\}&=1-F(x) \\[3ex]
P\{X\le a\}&=F(a) \\[1ex]
P\{X\lt a\}&=F(a-0) \\[1ex]
P\{X= a\}&=P\{X\le a\}-P\{X\lt a\}=F(a)-F(a-0) \\[3ex]
P\{a\lt X\lt b\}&=P\{X\lt b\}-P\{X\le a\}... \\[1ex]
P\{a\le X\le b\}&=P\{X\le b\}-P\{X\lt a\}... \\[3ex]
P\{a\lt X\le b\}&=P\{X\le b\}-P\{X\le a\}=F(b)-F(a) \\[1ex]
\end{split}\end{equation}
</script>
</p>
<h3 id="3">3. 反函数的概率密度函数</h3>
<p>定理：设随机变量具有概率密度 <script type="math/tex"> f_X(x),-\infty\lt x\lt\infty </script> ，又设函数 <script type="math/tex"> y = g(x) </script> 处处可导，且恒有 <script type="math/tex"> g'(x)\lt0 </script> 或 <script type="math/tex"> g'(x)\gt0 </script> 则 <script type="math/tex"> Y=g(X) </script> 是连续型随机变量，其概率密度为：<br />
<script type="math/tex; mode=display">
f_Y(y)=
\begin{cases}
f_X\{h(y)\}*|h'(y)| &\alpha\lt y\lt\beta\\[2ex]
0 &其他
\end{cases}
</script>
<br />
其中 <script type="math/tex"> \alpha=min\{g(-\infty),g(\infty)\},\beta=max\{g(-\infty),g(\infty)\} </script> ， <script type="math/tex"> x=h(y) </script> 为 <script type="math/tex"> y=g(x) </script> 的反函数。</p>
<h2 id="4_1">第4讲 多维随机变量及其分布</h2>
<h3 id="1-fxy">1. 联合概率密度函数 <script type="math/tex">f(x,y)</script>
</h3>
<p>
<script type="math/tex; mode=display">
f(x,y)
</script>
</p>
<h3 id="2-f_xxf_yy">2. 边缘概率密度函数 <script type="math/tex">f_X(x),f_Y(y)</script>
</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
推导F_X(x) &= \int_{-\infty}^x\bigg[\int_{-\infty}^{+\infty} f(x,y)dy\bigg]dx \\[1ex]
&= \int_{-\infty}^x\bigg[f_X(x)\bigg]dx\\[2em]
f_X(x) &= \int_{-\infty}^{\infty}f(x,y)dy \\[1ex]
f_Y(y) &= \int_{-\infty}^{\infty}f(x,y)dx
\end{split}\end{equation}
</script>
</p>
<h3 id="3-fxy">3. 联合分布函数 <script type="math/tex">F(x,y)</script>
</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
F(x,y) &= P\{X\le x, Y\le y\} \\[2ex]
&= \int_{-\infty}^y \int_{-\infty}^x f(x,y)dxdy  \\[2ex]
&= \int_{-\infty}^x \int_{-\infty}^y f(x,y)dydx
\end{split}\end{equation}
</script>
</p>
<p>公式<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&P\{x_1\lt X\le x_2,\ \ y_1\lt Y\le y_2\} \\[2ex]
&=F(x_2,y_2)-F(x_1,y_2)-F(x_2,y_1)+F(x_1,y_1)
\end{split}\end{equation}
</script>
</p>
<h3 id="4-f_xxf_yy">4. 边缘分布函数 <script type="math/tex">F_X(x),F_Y(y)</script>
</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
F_X(x)&=P\{X\le x\}=P\{X\le x, Y\lt+\infty\}=F(x,+\infty) \\[2ex]
F_Y(y)&=P\{Y\le y\}=P\{X\lt+\infty, Y\le y\}=F(+\infty,y)
\end{split}\end{equation}
</script>
</p>
<h3 id="5_1">5. 常见组合随机变量的概率密度函数</h3>
<h4 id="1-zxy">（1）<script type="math/tex"> Z=X+Y </script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f_{X+Y}(z)&=\int_{-\infty}^{\infty}f(x,z-x)dx\overset{独立}{=}\int_{-\infty}^{\infty}f_X(x)f_Y(z-x)dx \\[1em]
f_{X+Y}(z)&=\int_{-\infty}^{\infty}f(z-y,y)dy\overset{独立}{=}\int_{-\infty}^{\infty}f_X(z-y)f_Y(y)dy
\end{split}\end{equation}
</script>
</p>
<h4 id="2-zcfracyxzxy">（2）<script type="math/tex"> Z=\cfrac{Y}{X}、Z=XY </script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f_{\frac{Y}{X}}(z)&=\int_{-\infty}^{\infty}|x|f(x,zx)dx\overset{独立}{=}\int_{-\infty}^{\infty}|x|f_X(x)f_Y(zx)dx \\[1em]
         f_{XY}(z)&=\int_{-\infty}^{\infty}\cfrac{1}{|x|}f(x,\frac{z}{x})dx\overset{独立}{=}\int_{-\infty}^{\infty}\cfrac{1}{|x|}f_X(x)f_Y(\cfrac{z}{x})dx
\end{split}\end{equation}
</script>
</p>
<h4 id="3-zminxyzmaxxy">（3）<script type="math/tex"> Z=min\{X,Y\}、Z=max\{X,Y\} </script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
          Z &= min\{X,Y\} \\[2ex]
P\{Z\le z\} &= 1-P\{M\gt z\} \\[1ex]
            &= 1-P\{X\gt z,Y\gt z\} \\[1ex]
            &= F_X(z)+F_Y(z)-F(z,z) \\[1ex]
            &\overset{独立}{=} F_X(z)+F_Y(z)-F_X(z)F_Y(z) \\[1ex]
            &\overset{独立}{=} 1-[1-F_X(z)][1-F_Y(z)] \\[2em]
          Z &= max\{X,Y\} \\[2ex]
P\{Z\le z\} &= P\{X\le z,Y\le z\} \\[1ex]
            &= F(z,z) \\[1ex]
            &\overset{独立}{=} F_X(z)F_Y(z)
\end{split}\end{equation}
</script>
</p>
<h4 id="4_2">（4）叠加正态分布</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X,Y相互独立,X\sim N(\mu_1,\sigma_1^2),Y\sim N(\mu_2,\sigma_2^2) \\[2ex]
Z=X+Y,则Z\sim N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)
\end{split}\end{equation}
</script>
</p>
<h4 id="5_2">（5）二维正态分布</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(X,Y)&\sim N(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho) \\[2ex]
f_X(x)&=\cfrac{1}{\sqrt{2\pi}\sigma_1}e^{-\cfrac{(x-\mu_1)^2}{2\sigma_1^2}} \\[1ex]
f_Y(y)&=\cfrac{1}{\sqrt{2\pi}\sigma_2}e^{-\cfrac{(y-\mu_2)^2}{2\sigma_2^2}} 
\end{split}\end{equation}
</script>
</p>
<h2 id="6_1">第6讲 数字特征</h2>
<h3 id="1_3">1. 数学期望（均值）</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
E(Y) &= E[g(X)] = \int_{-\infty}^{\infty}g(x)f(x)dx \\[1ex] 
E(Z) &= E[g(X,Y)] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x,y)f(x,y)dxdy
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
  E(C) &= C \\[1ex]
 E(CX) &= CE(X) \\[1ex]
E(X+Y) &= E(X)+E(Y) \\[2ex]
若X和Y独立\rightarrow E(XY) &= E(X)E(Y) 
\end{split}\end{equation}
</script>
</p>
<h3 id="2_3">2. 方差</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
D(X) &= Var(X) \\[1ex]
&= E\{[X-E(X)]^2\} \\[1ex]
&= \int_{-\infty}^{\infty}[X-E(X)]^2f(x)dx \\[1ex]
&= E(X^2) - E[2XE(x)] + E[E(X)^2] \\[1ex]
&= E(X^2) - E(X)^2
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
  D(C) &= 0 \\[1ex]
D(X+C) &= D(X) \\[1ex]
 D(CX) &= C^2D(X) \\[1ex]
D(AX+BY) &= A^2D(X)+B^2D(Y) + \underline{2ABCov(X,Y)}
\end{split}\end{equation}
</script>
</p>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th align="left">分布</th>
<th>名称</th>
<th>概率密度</th>
<th align="center">
<script type="math/tex"> E(x) </script>
</th>
<th align="center">
<script type="math/tex"> D(x) </script>
</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">
<script type="math/tex"> X\sim (1,p) </script>
</td>
<td>0-1分布</td>
<td></td>
<td align="center">
<script type="math/tex"> p </script>
</td>
<td align="center">
<script type="math/tex"> (1-p)p </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim b(n,p) </script>
</td>
<td>两点分布</td>
<td></td>
<td align="center">
<script type="math/tex"> np </script>
</td>
<td align="center">
<script type="math/tex"> n(1-p)p </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim \pi(\lambda) </script>
</td>
<td>泊松分布</td>
<td>
<script type="math/tex"> p{\{x=k\}}=\cfrac{\lambda^ke^{-\lambda}}{k!} </script>
</td>
<td align="center">
<script type="math/tex"> \lambda </script>
</td>
<td align="center">
<script type="math/tex"> \lambda </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim U(a,b) </script>
</td>
<td>均匀分布</td>
<td>
<script type="math/tex"> f(x)=\cfrac{1}{b-a} </script> ，注意（ <script type="math/tex"> a\le x\le b </script> ）</td>
<td align="center">
<script type="math/tex"> \cfrac{a+b}{2} </script>
</td>
<td align="center">
<script type="math/tex"> \cfrac{(b-a)^2}{12} </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim Exp(\lambda) </script>
</td>
<td>指数分布</td>
<td>
<script type="math/tex"> f(x)=\lambda e^{-\lambda x} </script> ，注意（ <script type="math/tex"> x\gt0 </script> ）</td>
<td align="center">
<script type="math/tex"> \cfrac{1}{\lambda} </script>
</td>
<td align="center">
<script type="math/tex"> \cfrac{1}{\lambda^2} </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim N(\mu,\sigma^2) </script>
</td>
<td>正态高斯分布</td>
<td>
<script type="math/tex"> f(x)=\cfrac{1}{\sqrt{2\pi}\sigma}e^{-\cfrac{(x-\mu)^2}{2\sigma^2}} </script>
</td>
<td align="center">
<script type="math/tex"> \mu </script>
</td>
<td align="center">
<script type="math/tex"> \sigma^2 </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex">X\sim \chi^2(n)</script>
</td>
<td></td>
<td></td>
<td align="center">
<script type="math/tex"> n </script>
</td>
<td align="center">
<script type="math/tex"> 2n </script>
</td>
</tr>
</tbody>
</table></div>
<h3 id="3_1">3. 边缘概率密度</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f_X(x) = \int_{-\infty}^{\infty}f(x,y)dy \\[1ex]
f_Y(y) = \int_{-\infty}^{\infty}f(x,y)dx
\end{split}\end{equation}
</script>
</p>
<h3 id="4_3">4. 协方差</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
Cov(X,Y) &= E\{(X-E(X))(Y-E(Y))\} \\[1ex]
         &= E(XY) - E(X)E(Y) \\[2em]
Cov(X,Y) &= \int_{-\infty}^{\infty}\int_{\infty}^{\infty}(x-E(X))(y-E(Y))f(x,y)dxdy \\[1ex]
         &= \int_{-\infty}^{\infty}\int_{\infty}^{\infty}xyf(x,y)dxdy - \int_{-\infty}^{\infty}xf_X(x)dx\int_{-\infty}^{\infty}yf_Y(y)dy
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
Cov(aX,bY) &= ab*Cov(X,Y) \\[1ex]
Cov(X_1+X_2,Y) &= Cov(X_1,Y) + Cov(X_2,Y)
\end{split}\end{equation}
</script>
</p>
<h3 id="5_3">5. 相关系数</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\rho_{XY} = Corr(X,Y) &= \cfrac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X和Y相互独立&\Rightarrow(\rho_{XY} = 0)X和Y不相关 \\
&\nLeftarrow
\end{split}\end{equation}
</script>
</p>
<h2 id="7_1">第7讲 大数定理与中心极限定理</h2>
<h3 id="1_4">1. 依概率收敛</h3>
<p>
<script type="math/tex; mode=display">
设随机变量\ X\ 和随机变量序列\ \{X_n\}，对任意的\ \epsilon>0\ 有 \\[3ex]
\lim_{n\to+\infty}P\bigg\{\bigg|X_n-X\bigg|\ge\epsilon\bigg\}=0 \\[1ex]
或者\lim_{n\to+\infty}P\bigg\{\bigg|X_n-X\bigg|<\epsilon\bigg\}=1 \\[3ex]
则称随机变量序列\ \{X_n\}\ \underline{依概率收敛于}随机变量\ X
</script>
</p>
<h3 id="2_4">2. 大数定理</h3>
<p>切比雪夫(Chebyshev)不等式</p>
<h4 id="1_5">（1）切比雪夫大数定律</h4>
<p>
<script type="math/tex; mode=display">
假设\ \{X_n\}\ 是①\underline{独立}的随机变量序列 \\[1ex]
如果②\underline{D(X_i)\ 存在且一致有上界} \\[1ex]
即对常数\ C，使\ D(X_i)\le C\ 对一切\ i\ge1\ 均成立 \\[3ex]
\cfrac{1}{n}\sum_{i=1}^n X_i\overset{P}{\to}\cfrac{1}{n}\sum_{i=1}^n E(X_i)
</script>
</p>
<h4 id="2bernoulli">（2）伯努利(Bernoulli)大数定律</h4>
<p>
<script type="math/tex; mode=display">
假设\ \mu_n\ 是\ n\ 重伯努利实验中\ A\ 发生的次数 \\[1ex]
在每次事件中发生的概率为\ p(0<p<1) \\[3ex]
\cfrac{\mu_n}{n}\overset{P}{\to} p
</script>
</p>
<h4 id="3khinchine">（3）辛钦(Khinchine)大数定律</h4>
<p>
<script type="math/tex; mode=display">
假设\ \{X_n\}\ 是①\underline{独立}②\underline{同分布}的随机变量序列 \\[1ex]
如果③\underline{E(X_i)=\mu\ 存在} \\[3ex]
\cfrac{1}{n}\sum_{i=1}^n X_i\overset{P}{\to}\mu
</script>
</p>
<h4 id="4_4">（4）大数定理的结论</h4>
<p>
<script type="math/tex; mode=display">
\cfrac{1}{n}\sum_{i=1}^n X_i\overset{P}{\to}E\bigg(\cfrac{1}{n}\sum_{i=1}^n X_i\bigg)
</script>
</p>
<h3 id="3_2">3. 中心极限定理</h3>
<h4 id="1-levy-lindberg">（1）列维-林德伯格(Levy-Lindberg)定理</h4>
<p>
<script type="math/tex; mode=display">
假设\{X_n\}是①\underline{独立}②\underline{同分布}的随机变量序列 \\[1ex]
如果E(X_i)=\mu，D(X_i)=\sigma^2存在 \\[1ex]
则对任意 x ，分布函数 \\[3ex]
当 n 很大时，\overline X\sim N(\mu,\cfrac{\sigma^2}{n}) \\[1ex]
F_n(x)=P\bigg\{\cfrac{\overline X-\mu}{\sigma / \sqrt{n}}\le x\bigg\}=\Phi(x) \\[2em]
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
D(\overline X) = \cfrac{1}{n}D(X)=\cfrac{\sigma^2}{n}
</script>
</p>
</blockquote>
<h4 id="2-de-moivre-laplace">（2）棣莫弗-拉普拉斯(De Moivre-Laplace)定理</h4>
<h2 id="8">第8讲 统计量及其分布</h2>
<blockquote class="content-quote">
<p>几个重要的结论：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
E(X) &= \mu \\[1ex] 
D(X) &= \sigma^2 \\[2em]
E(\overline X) &= E(\cfrac{1}{n}\sum_{i=1}^n X_i) = \cfrac{1}{n}\sum_{i=1}^n E(X_i) = E(X) \\[1ex]
D(\overline X) &= D(\cfrac{1}{n}\sum_{i=1}^n X_i) = \cfrac{1}{n^2}\sum_{i=1}^n D(X_i) = \cfrac{1}{n}D(X) \\[2em]
\rho_{XY} &= \cfrac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}} \\[1ex]
\rho_{XY} &= \cfrac{Cov(\overline X,\overline Y)}{\sqrt{D(\overline X)}\sqrt{D(\overline Y)}}
= \cfrac{Cov(\overline X,\overline Y)}{\sqrt{\cfrac{1}{n}D(X)}\sqrt{\cfrac{1}{n}D(Y)}}
= \cfrac{Cov(\overline X,\overline Y)}{\cfrac{1}{n}\sqrt{D(X)}\sqrt{D(Y)}} \\[1ex]
Cov(\overline X,\overline Y) &= \cfrac{1}{n}\rho_{XY}\sqrt{D(X)}\sqrt{D(Y)}\\[2em]
E(S^2) &= E(\cfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline X)^2) \\[1ex]
&= \cfrac{1}{n-1}\sum_{i=1}^{n}\bigg[E(X_i^2)-2E(X_i)E(\overline X)+E(\overline X^2)\bigg] \\[1ex]
&= \cfrac{n}{n-1}\bigg[E(X_i^2)-E(\overline X^2)\bigg] \\[1ex]
&= \cfrac{n}{n-1}\bigg[\bigg(D(X)-[E(X_i)]^2\bigg)-\bigg(\cfrac{1}{n}D(X)-[E(X)^2]\bigg)\bigg] \\[1ex]
&= \cfrac{n}{n-1}\cfrac{n-1}{n}D(X) \\[1ex]
&= D(X)
\end{split}\end{equation}
</script>
<br />
</p>
</blockquote>
<h3 id="1_6">1. 统计量</h3>
<h4 id="1_7">（1）样本均值</h4>
<p>
<script type="math/tex; mode=display">
\overline X=\cfrac{1}{n}\sum_{i=1}^n X_i
</script>
</p>
<h4 id="2_5">（2）样本方差</h4>
<p>
<script type="math/tex; mode=display">
S^2=\cfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline X)^2
</script>
</p>
<p>注意与方差的区别<br />
<script type="math/tex; mode=display">
\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2
</script>
</p>
<h4 id="3_3">（3）样本标准差</h4>
<p>
<script type="math/tex; mode=display">
S=\sqrt{\cfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline X)^2}
</script>
</p>
<h4 id="4-k">（4）样本 <script type="math/tex"> k </script> 阶原点距</h4>
<p>
<script type="math/tex; mode=display">
A_k=\cfrac{1}{n}\sum_{i=1}^{n}X_i^k
</script>
</p>
<h4 id="5-k">（5）样本 <script type="math/tex"> k </script> 阶中心距</h4>
<p>
<script type="math/tex; mode=display">
B_k=\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\overline X)^k
</script>
</p>
<blockquote class="content-quote">
<p>样本方差的推导过程<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
假设S^2&=\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\overline X)^2 \\[1ex]
E(S^2)&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\overline X)^2\bigg] \\[1ex]
&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}\bigg((X_i-\mu)-(\overline X-\mu)\bigg)^2\bigg] \\[1ex]
&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}\bigg((X_i-\mu)^2-2(X_i-\mu)(\overline X-\mu)+(\overline X-\mu)^2\bigg)\bigg] \\[1ex]
&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2-2(\overline X-\mu)\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\mu)+(\overline X-\mu)^2\bigg] \\[1ex]
&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2-2(\overline X-\mu)(\overline X-\mu)+(\overline X-\mu)^2\bigg] \\[1ex]
&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2-(\overline X-\mu)^2\bigg] \\[1ex]
&=E\bigg[\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2\bigg]-E\bigg[(\overline X-\mu)^2\bigg] \\[1ex]
&=E(\sigma^2)-E\bigg[(\overline X-\mu)^2\bigg] \\[1ex]
&=\sigma^2-E\bigg[(\overline X-\mu)^2\bigg] 
\end{split}\end{equation}
</script>
<br />
<script type="math/tex">X_i</script> 和 <script type="math/tex">X</script> 独立同分布<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
E\bigg[(\overline X-\mu)^2\bigg]&=E\bigg[\overline{X}^2-2\overline{X}\mu+\mu^2\bigg] \\[1ex]
&=E(\overline{X}^2)-2E(\overline{X})\mu+\mu^2 \\[1ex]
&=E(\overline{X}^2)-\mu^2 \\[1ex]
&=D(\overline{X})+[E(\overline{X})]^2-\mu^2 \\[1ex]
&=D(\overline{X}) \\[1ex]
&=\cfrac{1}{n}\sigma^2 \\[2em]
E(\overline{X})&=E\bigg(\cfrac{1}{n}\sum_{i=1}^n X_i\bigg)=\cfrac{1}{n}\sum_{i=1}^nE(X_i)=\mu\\[1ex]
D(\overline{X})&=D\bigg(\cfrac{1}{n}\sum_{i=1}^n X_i\bigg)=\cfrac{1}{n^2}\sum_{i=1}^nD(X_i)=\cfrac{1}{n}\sigma^2 \\[1ex]
\end{split}\end{equation}
</script>
<br />
</p>
</blockquote>
<h3 id="2_6">2. 统计量四大分布</h3>
<h4 id="1nmusigma">（1）<script type="math/tex">N(\mu,\sigma)</script> 正态分布</h4>
<h4 id="2-chi2">（2） <script type="math/tex"> \chi^2 </script> 分布</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X_1,X_x,...,X_n&\sim N(0,1) \\[1em]
\Rightarrow X = \sum_{i=1}^{n}X_i^2&\sim\chi^2(n) \\[1em]
E(X) &= n \\[1ex]
D(X) &= 2n
\end{split}\end{equation}
</script>
</p>
<h4 id="3-t">（3） <script type="math/tex"> t </script> 学生分布</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X &\sim N(0,1) \\
Y &\sim\chi^2(n),\ X和Y独立 \\[1em]
\Rightarrow t = \cfrac{X}{\sqrt{\cfrac{Y}{n}}}&\sim t(n) \\[1em]
t_{1-\alpha}(n) &= -t_{\alpha}(n)
\end{split}\end{equation}
</script>
</p>
<h4 id="4-f">（4） <script type="math/tex"> F </script> 分布</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X &\sim\chi^2(n_1) \\
Y &\sim\chi^2(n_2),\ X和Y独立 \\[1em]
\Rightarrow F = \cfrac{X/n_1}{Y/n_2}&\sim F(n_1,n_2) \\[1em]
\cfrac{1}{F} &\sim F(n_2,n_1) \\[1ex]
F_{1-\alpha}(n_1,n_2) &\sim \cfrac{1}{F_\alpha(n_2,n_1)}
\end{split}\end{equation}
</script>
</p>
<h2 id="9">第9讲 参数估计</h2>
<h3 id="1_8">1. 矩估计</h3>
<ol>
<li>利用 <script type="math/tex"> \overline X=E(x) </script>
</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
      E(x)&=\int_{-\infty}^{\infty}xf(x;\theta)dx\\[1ex]
      E(x)&=\overline X=\cfrac{1}{n}\sum_{i=1}^n X_i\\[2ex]
\hat\theta&=f(\sum_{i=1}^n X_i)
\end{split}\end{equation}
</script>
</p>
<ol start="2">
<li>利用 <script type="math/tex"> \cfrac{1}{n}\sum X_i^2=E(x^2) </script>
</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
      E(x^2)&=\int_{-\infty}^{\infty}x^2f(x;\theta)dx\\[1ex]
      E(x^2)&=\cfrac{1}{n}\sum_{i=1}^n X_i^2\\[2ex]
\hat\theta&=f(\sum_{i=1}^n X_i^2)
\end{split}\end{equation}
</script>
</p>
<h3 id="2_7">2. 最大似然估计</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
                           L(\theta)&=\prod_{i=1}^nf(t_i;\theta)\\[1ex]
                      \ln{L(\theta)}&=\ln{\prod_{i=1}^nf(t_i;\theta)}\\[1ex]
\cfrac{d\{\ln{L(\theta)\}}}{d\theta}&=\cfrac{d\{\ln{\prod_{i=1}^nf(t_i;\theta)}\}}{d\theta} = 0\\[1ex]
                          \hat\theta&=\theta=f(\sum X_i)
\end{split}\end{equation}
</script>
</p>
<p>(2015年数学一)、若 <script type="math/tex"> g(n,\theta)=\cfrac{d\{\ln{L{\theta}\}}}{d\theta} </script> 为不含有 <script type="math/tex"> X_i </script> 的式子，当 <script type="math/tex"> g(n,\theta) </script> 随 <script type="math/tex"> \theta </script> 增大而增大时， <script type="math/tex"> \hat\theta=min\{X_1,X_2,...,X_n\} </script>
</p>
<h3 id="3_4">3. 区间估计</h3>
<p><strong>置信区间：</strong>设总体 <script type="math/tex"> X </script> 的分布函数 <script type="math/tex"> F(x;\theta) </script> 含有一个未知参数 <script type="math/tex"> \theta </script> ， <script type="math/tex"> \theta\in\Theta </script> （ <script type="math/tex"> \Theta </script> 是 <script type="math/tex"> \theta </script> 的可能取值范围），对于给定值 <script type="math/tex"> \alpha(0\lt\alpha\lt1) </script> ，若由来自 <script type="math/tex"> X </script> 的样本 <script type="math/tex"> X_1,X_2,...,X_n </script> 确定两个统计量 <script type="math/tex"> \underline\theta=\underline\theta(X_1,X_2,...X_n) </script> ， <script type="math/tex"> \overline\theta=\overline\theta(X_1,X_2,...X_n) </script> （ <script type="math/tex"> \underline\theta\lt\overline\theta </script> ），对于任意 <script type="math/tex"> \theta\in\Theta </script> ，满足<br />
<script type="math/tex; mode=display">
P\{\underline\theta(X_1,X_2,...X_n)\lt\theta\lt\overline\theta(X_1,X_2,...X_n\} \ge 1-\alpha
</script>
</p>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>项目</th>
<th>称为</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<script type="math/tex"> (\underline\theta,\overline\theta) </script>
</td>
<td>
<script type="math/tex"> \theta </script> 的置信水平为 <script type="math/tex"> 1-\alpha </script> 的<strong>置信区间</strong></td>
</tr>
<tr>
<td>
<script type="math/tex"> \underline\theta </script>
</td>
<td>
<script type="math/tex"> \theta </script> 的置信水平为 <script type="math/tex"> 1-\alpha </script> 的双侧置信区间的<strong>置信下限</strong></td>
</tr>
<tr>
<td>
<script type="math/tex"> \overline\theta </script>
</td>
<td>
<script type="math/tex"> \theta </script> 的置信水平为 <script type="math/tex"> 1-\alpha </script> 的双侧置信区间的<strong>置信上限</strong></td>
</tr>
<tr>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td><strong>置信水品</strong></td>
</tr>
</tbody>
</table></div>
<h3 id="4_5">4. 正态总体均值与方差的区间估计</h3>
<h5 id="nmusigma2">单个总体 <script type="math/tex"> N(\mu,\sigma^2) </script>
</h5>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th></th>
<th>置信水平</th>
<th>置信区间</th>
<th>置信区间</th>
</tr>
</thead>
<tbody>
<tr>
<td>均值 <script type="math/tex"> \mu </script> 的置信区间</td>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td>
<script type="math/tex"> \sigma^2 </script> 已知： <script type="math/tex"> (\overline X\pm\cfrac{\sigma}{\sqrt{n}}z_{\alpha/2}) </script>
</td>
<td>
<script type="math/tex"> \sigma^2 </script> 未知： <script type="math/tex"> (\overline X\pm\cfrac{S}{\sqrt{n}}t_{\alpha/2}(n-1)) </script>
</td>
</tr>
<tr>
<td>方差 <script type="math/tex"> \sigma^2 </script> 的置信区间</td>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td></td>
<td>
<script type="math/tex"> \mu </script> 未知： <script type="math/tex"> (\cfrac{(n-1)S^2}{\chi_{\alpha/2}^2(n-1)},\cfrac{(n-1)S^2}{\chi_{1-\alpha/2}^2(n-1)}) </script>
</td>
</tr>
<tr>
<td>标准差 <script type="math/tex"> \sigma </script> 的置信区间</td>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td></td>
<td>
<script type="math/tex"> \mu </script> 未知： <script type="math/tex"> (\cfrac{\sqrt{n-1}S}{\sqrt{\chi_{\alpha/2}^2(n-1)}},\cfrac{\sqrt{n-1}S}{\sqrt{\chi_{1-\alpha/2}^2(n-1)}}) </script>
</td>
</tr>
</tbody>
</table></div>
<h3 id="5_4">5. 估计量的评选标准</h3>
<h4 id="1_9">（1）无偏性</h4>
<p>
<script type="math/tex; mode=display">
若估计量\ \hat\theta=\hat\theta(X_1,X_2,...X_n)\ 的数学期望\ E(\hat\theta)\ 存在 \\[1ex]
对于任意\ \theta\in\Theta，有\\[4ex]
\underline{E(\hat\theta)=\theta} \\[4ex]
则称\ \hat\theta\ 是\ \theta\ 的无偏估计量
</script>
<br />
例如：<script type="math/tex"> E(\overline X)=\mu,E(S^2)=\sigma^2 </script> ，则对于任意分布</p>
<ul>
<li>样本均值 <script type="math/tex"> \overline X </script> 是总体均值 <script type="math/tex"> \mu </script> 的无偏估计；</li>
<li>样本方差 <script type="math/tex"> S^2=\cfrac{1}{n-1}\sum_{i=1}^n(X_i-\overline X)^2 </script> 是总体方差 <script type="math/tex"> \sigma^2 </script> 的无偏估计。</li>
</ul>
<h4 id="2_8">（2）有效性</h4>
<p>
<script type="math/tex; mode=display">
设\ \hat\theta_1=\hat\theta_1(X_1,X_2,...X_n)\\[1ex] 与\ \hat\theta_2=\hat\theta_2(X_1,X_2,...X_n)\ 都是\ \theta\ 的无偏估计量 \\[1ex]
对于任意\ \theta\in\Theta ，有 \\[4ex]
D(\hat\theta_1)\le D(\hat\theta_2) \\[4ex]
且至少对于某一个\ \theta\ 上式等号不成立，则称\ \hat\theta_1\ 较\ \hat\theta_2\ 有效
</script>
</p>
<h4 id="3_5">（3）相合性</h4>
<p>
<script type="math/tex; mode=display">
设\ \hat\theta(X_1,X_2,...X_n)\ 为参数\ \theta\ 的估计量 \\[1ex]
对于任意\ \theta\in\Theta，任意\ \epsilon\gt0 ，有 \\[4ex]
\underline{\lim_{n\to\infty}P\{|\hat\theta-\theta|\lt\epsilon\}=1} \\[4ex]
则称\ \hat\theta\ 为\ \theta\ 的相合估计量
</script>
</p>
<blockquote class="content-quote">
<p>【2021.9】<br />
<script type="math/tex; mode=display">
设\ (X_1,Y_1),(X_2,Y_2)...(X_n,Y_n)\\[1ex] 为来自总体\ N(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho)\ 的简单随机样本 \\[1ex]
令\ \theta=\mu_1-\mu_2,\ \theta=\overline X-\overline Y
</script>
<br />
解答<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
E(\theta) 
&=E(\overline X-\overline Y) \\
&=E(\overline X)-E(\overline Y) \\
&=\mu_1-\mu_2 \\[1ex]
&\theta\ 是\ \theta\ 的无偏估计 \\[1em]
D(\theta)
&=D(\overline X-\overline Y) \\
&=D(\overline X)+D(\overline Y)-2Cov(\overline X,\overline Y) \\
&=\cfrac{1}{n}\sigma_1^2+\cfrac{1}{n}\sigma_2^2-2\cfrac{1}{n}\rho\sigma_1\sigma_2 \\
&=\cfrac{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2}{n}
\end{split}\end{equation}
</script>
<br />
</p>
</blockquote>
<h3 id="6_2">6. 假设检验</h3>
<p>
<script type="math/tex">H_0</script> ：研究者想收集证据予以 <strong>反对</strong> 的假设，一般带等号</p>
<p>
<script type="math/tex">H_1</script> ：研究者想收集证据予以 <strong>支持</strong> 的假设，一般不带等号</p>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>项目</th>
<th>没有拒绝 <script type="math/tex">H_0</script>
</th>
<th>拒绝 <script type="math/tex">H_0</script>
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<script type="math/tex">H_0</script> 真</td>
<td>正确决策（<script type="math/tex">1-\alpha</script>）</td>
<td>第一类错误，弃真错误（<script type="math/tex">\alpha</script>）</td>
</tr>
<tr>
<td>
<script type="math/tex">H_0</script> 伪</td>
<td>第二类错误，取伪错误（<script type="math/tex">\beta</script>）</td>
<td>正确决策（<script type="math/tex">1-\beta</script>）</td>
</tr>
</tbody>
</table></div>
<blockquote class="content-quote">
<p>【2021.10】<br />
<script type="math/tex; mode=display">
设\ X_1,X_2...X_{16}\ 是来自总体\ N(\mu,4)\ 的简单随机样本 \\[1ex]
考虑假设检验问题：\\[1ex]
H_0:\mu\le10,\ H_1:\mu\gt10 \\[1ex]
若该检验问题的拒绝域为\ W=\{\overline X\gt11\},其中\ \overline X=\cfrac{1}{16}\sum_{i=1}^{16}X_i \\[1ex]
当\ \mu=11.5\ 时，该检验问题犯第二类错误的概率为
</script>
<br />
解答<br />
<script type="math/tex; mode=display">
犯第二类错误的概率为\ P\{\overline X\le11\} \\[1ex]
E(\overline X)=E(X)=\mu=11.5 \\[1ex]
D(\overline X)=\cfrac{1}{n}D(X)=\cfrac{1}{16}D(X)=\cfrac{1}{16}·4=\cfrac{1}{4} \\[1ex]
\overline X\sim N(11.5,\cfrac{1}{4}) \\[1ex]
P\{\overline X\le11\}=P\{\cfrac{\overline X-11.5}{1/2}\le\cfrac{11-11.5}{1/2}\}
=\Phi(-1)=1-\Phi(1)
</script>
<br />
</p>
</blockquote>
<h3 id="_2"></h3>
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